More generally, whenever the underlying-set functor V(I,−):V→SetV(I,-) : V\to Set is conservative, since the morphism of hom-objects C(X,YZ)→C(X×Y,Z)C(X,Y^Z) \to C(X\times Y,Z) induced by the evaluation morphism YZ×Y→ZY^Z\times Y \to Z has invertible image in SetSet, hence is itself invertible if V(I,−)V(I,-) is conservative.

When VV is cartesian monoidal and C=VC=V: a cartesian closed category is automatically enriched-cartesian-closed over itself. In other words, the defining isomorphisms V0(X×Y,Z)≅V0(X,ZY)V_0(X\times Y,Z) \cong V_0(X, Z^Y) induce, by the Yoneda lemma, isomorphisms of exponential objectsZX×Y≅(ZY)XZ^{X\times Y} \cong (Z^Y)^X.